Unequal Temperaments (DeMorgan)

[Richard Moody]

----- Original Message -----
From: <A440A@AOL.COM>
To: <pianotech@ptg.org>
Sent: Sunday, June 11, 2000 12:06 PM
Subject: Re: Unequal Temperaments (DeMorgan)


> Ric asks:
> << What is a DeMorgan?

Ed Replies.....
>    It is characterized by fifths ranging from 698.5 cents (F#-C#) to 700.5
> cents (C-G)   .  The thirds vary, in opposition to the fifths, from 396
cents
> to 404 cents.

Ric----
    Well, , , this is  getting closer.   If smallest third is 396 that is
rather sharp of pure which is 386.     Of course ET 3rd is 400 so 404 is
sharp of ET.  All of the thirds will beat---some slower than ET, some
faster. I would suspect a system of some fifths pure and some fifths flat.
Pure fifth is 702..   Now the riddle here for the aural tuner is how the
instructions read.  For the ETD users there must be a cents table.   So for
DeMorgan no pure thirds, but no wolf  5th. His instructions would be
interesting to read to see how he does this.   If he had stayed with 700.5
for fifths he would have close to  ET as it can be argued that the
inharmonicity of the octave might require fifths a tad wider than
theoritical ET,(700) as in 700.5 *12/7 =  1200.857.   (as 12 5ths gives an
octave but 7 octaves up)  Now an octave of 1200.857 starting at A3 220  is
220*2^(cents/1200)   or 220*2^(1200.857/1200) =  440.22.   Hmmm this could
very well be what we tune at in reality.
    From all of this it looks like the DeMorgan is awfully close to ET. It
would now be interesting to read his instructions to see what  he is getting
at. Or since cents are given for some of the notes, (which I assume come
from the interpretation of his instructions) I wonder what all of the notes
in cents would look like on a spread sheet?    ---ric



    This is a temperament by August DeMorgan, in the 1840's.  Jorgensen
> documents it in "Tuning".  The major difference between it and all the
well
> temperaments is that the order of increasing dissonance in the tonic
thirds
> is reversed.  Whereas virtually every temperament proposed between
> Werckmeister and Young had the ascension of dissonance beginning at C, (or
in
> the case of Valotti, F)  and following the circle of fifths.  The DeMorgan
> tuning reverses that, so that the key  with more accidentals has a tonic
> third that is closer to Just than one with fewer.  Chopin's music, with so
> many sharps etc,  is strongly affected by this tuning.  It is listed as a
> quasi-equal temperament.
>    It is characterized by fifths ranging from 698.5 cents (F#-C#) to 700.5
> cents (C-G)   .  The thirds vary, in opposition to the fifths, from 396
cents
> to 404 cents.
>
> Regards,
> Ed